re: Backing out of Purchase Agreement

Screw them. They signed the agreement and knew or should have known the contents of said agreement. It's right there in black and white.

They found an "issue" with the AC that they didn't even notify you of in proper manner and you even offered to fix the made up "issue" by replacing the AC. I know in LA that's all that's required for you to do is to fix any issue they found. The fact that they aren't OK with you fixing the issue they (incorrectly) notified you of means they are acting in bad faith. Go after them legally just as its laid out in the contract that they signed. 10% penalty, specific performance, whatever. Again, they are the ones acting in bad faith, not you.